Prediction from Quasi-Random Time Series
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Prediction from Quasi-Random Time Series
Lorenza SaittaDipartimento di Informatica
Università del Piemonte OrientaleAlessandria, Italy
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The Problem
• Data
– ORACLE DataBase structure:
< Client, #Order, Product, Quantity, Value, Lit/Euro >
Each order may contain more than one product, or the same product with different quantities and prices
– Time series representing the daily sells of 16,265 products, grouped into 1,004 categories
• Goal
– From years 1999, 2000, 2001 --> Predict year 2002
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Pre-Processing
• Impossible to handle each product or each category separately --> Necessity to group/select products/categories
• Impossible to handle groups of products/categories for each client --> Necessity to group clients
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Product Selection• For each year :
– Order products according to decreasing revenue sharing
– Order products according to decreasing sold quantity – Select select products that globally cover 80% of
revenue (sold quantity)
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Grouping Products by Revenue Sharing
• Selection of those products that globally cover 80 % of revenue -> 53 Products
{3668995786, 9954462001, 5.3693}
{3179208029, 9950062610, 10.0219}
{2908553900, 9953071610, 14.2783}
{2833938896, 9950242825, 18.4256} { Value, Code, Cumulative function}
{2357611762, 9951252610, 21.8758}
{2111514418, 9952991610, 24.9659}
{2029888070, 9953091620, 27.9365}
…………
{2331985486, 9956521850, 79.6994}
{2325944507, 9952742830, 80.1763}
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Annual Data
• Grouping by “days” or “weeks” produces apparently random time series
• Grouping starts being meaningful at the “month” level
• Necessity of correct alignment among years
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Yearly Comparison
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Mese
2542905 = Bagno Fe. Nat. Idratante, 500 Ml,4x3 Pz2001
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Sequence over 31 Months
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2542905 = Bagno Fe. Nat. Idratante, 500 Ml, 4x3 Pz Quantità
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Overview of Analysis
• Single Products– General statistical characteristics
– Yearly analysis (monthly, trimester data)
– Three-year sequences
– Time series analysis (trend, periodicity, oscillations, noise)• Additive Model Yt = Tt + Ct + St + Rt
• Multiplicative Model Yt = Tt * Ct * St * Rt
– Model of the selling process
• Products hierarchies
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General Observation
• There is no substantial difference in behavior between the single and aggregated product analysis
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Statistical Analysis
• Autocorrelation Coefficient (ACF)95% Significativity Plot (No significant correlations)
• Partial Autocorrelation up to 6 points (PACF) 95% Significativity Plot
(No significant correlations)
• Smoothing (Simple moving average, Spencer's and Henderson's weighted moving averages, EWMA, 3RSS)
(Series are too short)
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Statistical Analysis
• Periodgram (Fourier Analysis )(No clear seasonality)
• Randomness Test (Runs above and below median, Runs up and down, Box-Pierce test)
(All serie pass every test)
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Series Decomposition• Trend, Seasonality, Cyclicity, Noise
– Additive Model Yt = Tt + Ct + St + Rt
– Multiplicative Model Yt = Tt * Ct * St * Rt
• Decomposition– Smooth the data using a moving average of length equal to the length of
seasonality. This estimates the trend-cycle.– Divide the data by the moving average (if using the multiplicative method) or
subtract the moving average from the data (if using the additive method). This estimates the seasonality.
– Average the results for each season separately and rescale so that an average month equals 100 (multiplicative) or 0 (additive). This gives the seasonal indices.
– Adjust the data for the estimated trend-cycle and seasonality, yielding the irregular or residual component.
– Divide the original data values by the appropriate seasonal index if using the multiplicative method, or subtract the index if using the additive method. This gives the seasonally adjusted data.
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Qu
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10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Mesi
0032600 = Talco Busta F.A.,50 gr, 4x12 Pz Quantità
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Trimester Analysis • By aggregating data by trimesters, a more
stable behavior emerges
• Eight typical patterns
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Trimester Analysis
• All relevant variables values are compared on the trimester level, year by year
• A unique sequence is formed for better approximability
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Cumulative Analysis by Year
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Global Time Series
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Comparison on Cumulative Graphs
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Comparison on Differential Graphs
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True
Automatic prediction
Manual prediction
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Model of the Selling Process
Qt Vt
pt
xt
t
Q = Target quantity to buy
Qt = Bought quantity up to t
Vt = Expenditure up to t
t = Missing quantity
pt = Offered price at t
xt = Quantity to buy at t
Parameters
Qt x i ri i1
t 1
Vt v ii1
t 1
t = Q - Qt
x tQ
121
A
pte
t
V t
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Prediction from Model
• Regression on Cumulative function
x ty(t) A
pte
t
Vt
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Comparison on Differential Graphs
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True
Automatic prediction
Manual prediction
Model prediction with running adjustment
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Conclusions
• Prediction below the monthly scale is impossible
• Prediction at the trimester scale is possible• Real time adjustments can be made if a two
months delay is acceptable• Results can be used as a Min-Max strategy
in Game Theory